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Oscillations of first order linear retarded differential equations. (English) Zbl 0731.34080
The author considers the first order retarded differential equation $(E)\quad x'(t)+\sum^{n}_{k=1}p_ k(t)x(t-\tau_ k(t))=0,$ where $$p_ k$$ and $$\tau_ k$$ $$(k=1,2,...,n)$$ are non-negative continuous functions on an interval $$[t_ 0,\infty)$$, and $$\lim_{t\to \infty}(t- \tau_ k(t))=\infty$$ $$(k=1,2,...,n)$$. Main results of the paper establish sufficient conditions for all solutions of (E) to be oscillatory and conditions, under which (E) has at least one positive solution x with $$\lim_{t\to \infty}x(t)=0$$ and such that $$x(t)\leq \exp \{-\lambda \int^{t}_{t_ 0}[\sum^{n}_{j=1}p_ j(s)]ds]$$ for all large t($$\lambda$$ is a positive number).
In the paper we find several relations with earlier known results.

##### MSC:
 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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